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Improved Ensemble Empirical Mode Decomposition and its
Applications to Gearbox Fault Signal Processing
Jinshan Lin
School of Mechatronics and Vehicle Engineering, Weifang University
Weifang, 261061, China
Abstract Ensemble empirical mode decomposition (EEMD) is a noise-
assisted method and also a significant improvement on empirical
mode decomposition (EMD). However, the EEMD method lacks
a guide to choosing the appropriate amplitude of added noise and
its computation efficiency is fairly low. To alleviate the
problems of the EEMD method, the improved complementary
EEMD method (ICEEMD) was proposed. Furthermore, the
ICEEMD method was used to analyze realistic gearbox faulty
signals. The results indicate that the ICEEMD method has some
advantages over the EEMD method in alleviating the mode
mixing and splitting as well as reducing the time cost and also
outperforms the CEEMD method in alleviating the mode mixing
and splitting. The paper also indicates that the ICEEMD method
seems to be an effective and efficient method for processing
gearbox fault signals.
Keywords: Complementary Ensemble Empirical Mode
Decomposition(CEEMD), Improved Complementary Ensemble
Empirical Mode Decomposition(ICEEMD), Gearbox, Signal
Processing.
1. Introduction
It is a challenging task to develop signal processing
techniques for non-stationary and noisy signals, which has
attracted considerable attentions recently [1]. Many
methods, such as short time frequency transform [2] and
wavelet transform [3] , have been proposed for solving the
problem and proved useful in some applications. However,
because these methods usually need a priori knowledge
about the researched signals, they naturally lack the self-
adaption for the researched signals. The Wigner-Ville
distribution has high time-frequency resolution, but its
cross terms is unbearable [4]. Empirical mode
decomposition (EMD) is a self-adaptive method and
suitable to analyzing the non-stationary and nonlinear
signals [4], which has been successfully applied to various
fields [4]. Nevertheless, when the EMD algorithm is used
to deal with a signal with intermittency, the mode mixing
often emerges as an annoying problem [5-7]. To overcome
the mode mixing problem, ensemble empirical mode
decomposition (EEMD) is presented in place of EMD [8].
The EEMD method adds some white noise with limited
amplitude to the researched signals, sufficiently taking
advantage of the statistical characteristics of white noise
whose energy density is uniformly distributed throughout
the frequency domain, then projects the signal components
onto the proper frequency bands and, finally, the added
white noise can been counteracted by ensemble mean of
enough corresponding components [8]. Therefore, the
EEMD method is considered as a significant improvement
on the EMD method and recommended as a substitute for
the EMD method [8]. Indeed, the EEMD method has
shown its superiority over the EMD method in some
applications [9].
However, the EEMD method lacks a guide to how to
choose the appropriate amplitude of the added noise and
its computation efficiency is fairly low. As a result, the
inappropriate amplitude of the added white noise for the
EEMD method is going to cause the mode mixing and
splitting that often exists in the EMD method [10, 11].
Although the reference [8] suggested that the amplitude of
the added white noise should be about 0.2 times of the
standard deviation of the investigated signal, unfortunately,
with the suggested value, the decomposition results from
the EEMD method often deviate from the realistic
contents of the signals [11]. In addition, to further remove
the residual of the added white noise and reduce a waste of
time for the EEMD method, the complementary ensemble
empirical mode decomposition (CEEMD) has been
addressed to replace the EEMD method as a standard
version of the EMD method [10]. Notwithstanding, the
CEEMD method only partly enhances the computation
efficiency of the EEMD method, and the above first
problem regarding the EEMD method still remains
untouched. Additionally, if the researched signal is a noisy
signal in itself, its intrinsic noise will inevitably interact
with the noise added through the EEMD method, which
may further complicate the above first problem regarding
the EEMD method. In particular, when the researched
signals become very noisy, the above first problem
regarding the EEMD method leaves a gap.
This paper explores the above two problems concerning
the EEMD method. Then, the improved CEEMD
(ICEEMD) method was proposed. Applications to analysis
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012 ISSN (Online): 1694-0814 www.IJCSI.org 194
Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.
of defective gearbox signals proved the superiority of the
ICEEMD method over the EEMD method.
2. The EMD and its Several Variations
2.1 The EMD method
The EMD method can self-adaptively decompose any a
non-stationary and nonlinear signal into a set of intrinsic
mode functions (IMFs) from high frequency to low
frequency, which may be written as
1
( ) ( ) ( )N
i N
i
x t c t r t
(1)
where ci(t) indicates the ith IMF and rN(t) represents the r
esidual of the signal x(t). An IMF is a function which mus
t satisfy the following two conditions: (1) the number of e
xtrema and the number of zero crossings either equal to e
ach other or differ at most by one, and (2) at any point, th
e local average of the upper envelope and the lower envel
ope is zero [8]. The residual rN(t) usually is a monotonic f
unction or a constant.
2.2 The Ensemble EMD
An annoying problem associated with the EMD method is
the mode mixing due to intermittency, defined as either a
single IMF consisting of widely disparate scales, or a
signal residing in different IMF components. To alleviate
the imperfection of the EMD method, the ensemble EMD
(EEMD), a noise-assisted method, is proposed. The
EEMD method can be stated as follows:
( ) ( ) ( ), 1,2, ,m mx t x t w t m N (2)
, ,
1
( ) ( ) ( ), 1,2, ,L
m m i m L
i
x t c t r t m N
(3)
, ,
1
( ) ( ) ( ), 1,2, ,L
m m i m L
i
x t c t r t m N
(4)
where x(t) is the original signal, wm(t) is the mth added
white noise, xm(t) is the noisy signal of the mth trial, cm,i(t)
is the ith IMF of the mth trial, L is the number of IMFs
from the EMD method, and N is the ensemble number of
the EEMD method.
The EEMD method adds white noise with the finite
amplitude to the signal, sufficiently taking advantage of
the uniform statistic characteristics of white noise in the
frequency domain, projects the different frequency signal
components onto the corresponding frequency banks and,
as a result, effectively overcomes the mode mixing due to
the existence of intermittency [8]. Nonetheless, to totally
clear the residual of the added white noise from the IMFs
of the EEMD method, a large ensemble number is usually
demanded, which will cause a tremendous waste of time.
2.3 The Complementary EEMD method
To better eliminate the residual of added white noise
persisting in the IMFs of the EEMD method and raise the
computation efficiency of the EEMD method, the
complementary ensemble EMD (CEEMD) [10], a novel
noise-enhanced method, is presented. The CEEMD
method adds white noise in pairs with one positive and
another negative to the original signal and then produces
two sets of ensemble IMFs. Hence, two different
combinations of the original signal and the added white
noise can be obtained, i.e.
( )1 1( ), 1,2, ,
( )1 1( )
m
mm
x tx tm N
w tx t
(5)
where x(t) is the original signal, wm(t) is the mth added
white noise, xm+(t) is the sum of the original x(t) and the
mth added white noise wm(t), and xm-(t) is the difference of
the original x(t) and the mth added white noise wm(t). In
light of (3) and (4), the original signal x(t) can be
expressed as
, ,
1 1
, ,
1
1( ) ( ) ( )
2
1( ) ( )
2
L N
m i m i
i m
N
m L m L
m
x t c t c tN
r t r tN
(6)
where c+m,i(t) is the ith IMF of xm
+(t) and c-m,i(t) is the ith
IMF of xm-(t). Although the CEEMD method appears to be
able to completely remove the residual of the added white
noise persisting in the IMFs of the EEMD method, it still
remains unsolved how to choose the appropriate amplitude
of the added white noise for the EEMD method.
3. The Improved CEEMD Method
3.1 The Choice of Amplitude of the Added Noise
The amplitude of the added white noise is a key parameter
of the EEMD method, which will exert a decisive impact
on whether or not the EEMD method can yield the
reasonable decomposition results. If the added noise is too
weak to bring the changes of extrema of the original signal,
the EEMD method will degenerate into the EMD method.
Conversely, if the added noise is too strong to reveal the
original signal, the EEMD method will derive meaningless
results which are mainly controlled by the added noise and
scarcely associated with the original signal [11], whether
or not the ensemble number is large enough. The reference
[11] demonstrated that the decomposition results of the
EEMD method varied with the different amplitude of the
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012 ISSN (Online): 1694-0814 www.IJCSI.org 195
Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.
added noise and considered it appropriate for the EEMD
method to set SNR in the range of 50-60 dB. In fact, for
the simple simulated example by which the conclusions
were drawn in [11], when the amplitude of the added noise
is set as 0.01, which is considered an optimal choice by
[11], the SNR between the original signal and the added
noise is approximately 37 dB which is outside the range of
50-60 dB. Accordingly, the conclusions given by [11],
relating to the choice of the amplitude of the added white
noise of the EEMD method, is not entirely dependable.
Then, the problem is further investigated in this paper
using two simulated signals. Thus, the Pearson’s
correlation coefficient (PCC) is used as a parameter to
measure the performance of the EEMD method with
different amplitude of the added white noise. Here, the
ensemble number of the EEMD method is set as 100.
First, a simple noiseless simulated signal was used to
examine the choice of the amplitude of the added white
noise for the EEMD method. The signal is a combination
of a low-frequency sinusoid component x1(t) and a high-
frequency damped transient component x2(t), shown in Fig.
1, and its formula can refer to [11]. The relationship
between the PCCs and the amplitude of the added white
noise is illustrated in Fig. 2. As shown in Fig. 2, when the
amplitude of the added white noise is 0.0063, the two
PCCs almost simultaneously reach their maximum values.
Table 1 exhibits the average powers of two components of
the signal and the square roots of the average powers. As
seen in Table 1, the value of 0.0063 just equals to the
square root of the average power of the weak transient
component x1(t). As a result, the square root of the average
power of the weak transient component apparently
approximates the optimal amplitude of the added white
noise.
-1
0
1
signal
-0.05
0
0.05
x1(t)
Am
plit
ude
0 0.005 0.01 0.015 0.02 0.025-1
0
1
x2(t)
Time(s)
Fig. 1 A simple noiseless simulated signal and two components.
0 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
1
1.2
The amplitude of the added white noise
Pears
on's
corr
ela
tion c
oeff
icie
nt
(PC
C)
PCC corresponding to x1
PCC corresponding to x2
0.0063
Fig. 2 The relationship between the Pearson’s correlation coefficient
(PCC) and the amplitude of the added white noise for the simple signal.
Table 1: The average powers of two components of the simple noiseless
simulated signal and the square roots of the average powers
Parameter
The two components of the
simple signal
1( )x t 2 ( )x t
The mean power 4.0106×10-5 0.5
The square root of the
mean power 0.0063 0.7071
Subsequently, a complex noiseless simulated signal was
utilized to further verify the conclusion. The signal
consisting of four components imitates realistic vibration
signals of a rolling bearing, shown in Fig. 3, and its
formula can refer to [11]. The relationship between the
PCCs and the amplitude of the added white noise is
illustrated in Fig. 4. As shown in Fig. 4, when the
amplitude of the added white noise lies in the range of
0.0085-0.0138, the four PCCs almost simultaneously
reach their maximum values. Table 2 depicts the average
powers of four components of the signal and the square
roots of the average powers. As shown in Table 2, the
value of 0.0085 is just equal to the square root of the
average power of the weak sinusoid component x4(t) and
the value of 0.0138 is just equal to the square root of the
average power of the weak transient component x1(t).
More generally, Fig. 4 indicates that an optimal internal
of the amplitude of the added white noise for the EEMD
method may lie between the square root of the average
power of the weak sinusoid component and the square
root of the average power of the weak transient
component, which is in accordance with the conclusions
drawn from the above simple example.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012 ISSN (Online): 1694-0814 www.IJCSI.org 196
Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.
-0.20
0.2signal
-0.10
0.1x1(t)
-0.050
0.05x2(t)
Am
plit
ude
-0.050
0.05x3(t)
0 0.01 0.02 0.03 0.04 0.05-0.02
00.02
x4(t)
Time(s)
Fig. 3 A complexly-simulated signal and its four components.
0 0.0085 0.02 0.04 0.06 0.08 0.10
0.2
0.4
0.6
0.8
1
1.2
The amplitufe of the added white noise
Pears
on's
corr
ela
tion c
oeff
icie
nt
(P
CC
)
PCC corresponding to x1PCC corresponding to x2PCC corresponding to x3PCC corresponding to x4
0.0138
Fig. 4 The relationship between the Pearson’s correlation coefficient
(PCC) and the amplitude of the added white noise in a complex noiseless
simulated signal.
Table 2: The average powers of four components of the complex
noiseless simulated signal and the square roots of the average powers
Parameter
The four components of the complex
signal
1( )x t 2 ( )x t 3 ( )x t 4 ( )x t
The mean power 1.9032 1.9377 5.4450 0.7200
The square root of the
mean power 0.0138 0.0139 0.0233 0.0085
3.2 The improved CEEMD
On the premise that the noise is neglected, the above
section gives an optimal interval of the amplitude of the
added white noise for the EEMD method. Actually, a
realistic signal is usually contaminated by strong or weak
noise. When the EEMD method is applied to a noisy
signal, the intrinsic noise will inevitably interact with the
added noise, which can make a great impact on how much
extrinsic noise should be added. Although the reference [8]
has suggested that the amplitude of the added white noise
should be about 0.2 times of the standard deviation of the
investigated signal, the conclusion is pretty rough and
inappropriate in many cases [11]. To alleviate the problem
existing in the EEMD method for analyzing a noisy signal,
the improved CEEMD (ICEEMD) is proposed. First, the
noisy signal is roughly decomposed using the CEEMD
algorithm. Then, both the weak transient component and
the weak sinusoid component are obtained. As a result, the
optimal interval of the amplitude of the added white noise
can be determined. In the end, with the amplitude of the
added white noise lying in the optimal interval obtained in
the previous step, the CEEMD method is again performed.
4. Experiment verification
To further assess its performance, the ICEEMD method
was exploited to examine the gearbox vibration data
provided by Kayvan J. Rafiee [12]. The gearbox was
running by the driving gear meshing with the driven one.
The rotation speed of the input shaft is 24.05Hz, the
rotation speed of the output shaft is 29.06Hz and the
meshing frequency is 697.5Hz [12]. The signals were
measured from the driving gear. The normal gearbox
signal and the broken-tooth gearbox signal are shown in
Fig. 5. Subsequently, the EEMD method, the CEEMD
method and the ICEEMD method were utilized to explore
the two signals, and the corresponding HHT spectra are
shown in Fig. 6, Fig. 7 and Fig. 8, respectively. As shown
in Fig. 6(a), Fig. 7(a) and Fig. 8(a), there are no obvious
periodic characteristics in the HHT spectra of the normal
gearbox signal; conversely, as shown in Fig. 6(b), Fig. 7(b)
and Fig. 8(b), there are obvious periodic characteristics in
the HHT spectra of the broken-tooth gearbox signal.
However, as seen in Fig. 6(b) and Fig. 7(b), none of other
explicit information can be found, in addition to the
frequency bands scattered nearly throughout the frequency
range only at the instant when the shocks happen, which
implies that there occurs the mode mixing or splitting in
the two methods because of the inappropriate amplitude of
the added noise. Instead, as seen in Fig. 8(b), in addition to
the frequency bands scattered nearly throughout the
frequency range only at the instant when the shocks
happen, there is another an instantaneous frequency curve
similar to a cosine curve (highlighted with the red curve)
with the modulation frequency of 24Hz and the carrier
frequency of about 4800Hz, where the frequency 24Hz
almost equals to the rotation speed of the input shaft and
the frequency 4800Hz approaches the twelve times of the
meshing frequency, which denotes that there occurs a
frequency modulation phenomenon in the broken-tooth
signal. Consequently, the comparisons between the three
HHT spectra from Fig. 6(b), Fig. 7(b) and Fig. 8(b) prove
that the ICEEMD method greatly alleviates the mode
mixing and splitting of the EEMD/CEEMD method and
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012 ISSN (Online): 1694-0814 www.IJCSI.org 197
Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.
can extract more and useful information from the faulty
signals, which is essential for fault diagnosis of gearboxes.
In addition, Fig. 9 presents that ICEEMD, comparable to
CEEMD, can reduce a waste of time by 80% compared
with the EEMD method. This indicates that the ICEEMD
method is seemingly an effective and efficient method for
gearbox fault signal processing.
0 0.05 0.1 0.15 0.2 0.25
-200
-100
0
100
200
Time(s)
Am
plit
ude(m
.s-2
)
a
0 0.05 0.1 0.15 0.2 0.25-600
-400
-200
0
200
400
600
Time(s)
Am
plit
ude(m
.s-2
)
b
Fig. 5 The two gearbox vibration signals: (a) The normal gearbox signal;
(b) The broken-tooth gearbox signal.
Time(s)
Fre
quency(k
Hz)
0.05 0.1 0.15 0.2 0.25
2
4
6
8
50
100
150
a
Time(s)
Fre
quency(k
Hz)
0.05 0.1 0.15 0.2 0.25
2
4
6
8
100
200
300
400
500b
Fig. 6 The HHT spectra of the two vibration signals using the EEMD
method: (a) The normal gearbox signal; (b) The broken-tooth gearbox
signal.
Time(s)
Fre
quency(k
Hz)
0.05 0.1 0.15 0.2 0.25
2
4
6
8
50
100
150
a
Time(s)
Fre
quency(k
Hz)
0.05 0.1 0.15 0.2 0.25
2
4
6
8
100
200
300
400
500b
Fig. 7 The HHT spectra of the two vibration signals using the CEEMD
method: (a) The normal gearbox signal; (b) The broken-tooth gearbox
signal.
Time(s)
Fre
quency(k
Hz)
0.05 0.1 0.15 0.2 0.25
2
4
6
8
50
100
150
a
Time(s)
Fre
quency(k
Hz)
0.05 0.1 0.15 0.2 0.25
2
4
6
8
100
200
300
400
500b
Fig. 8 The HHT spectra of the two vibration signals using the ICEEMD
method (The red cosine curve is added to highlight the instantaneous
frequency curve.): (a) The normal gearbox signal; (b) The broken-tooth
gearbox signal.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012 ISSN (Online): 1694-0814 www.IJCSI.org 198
Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.
EEMD CEEMD ICEEMD0
10
20
30
40
50
60
Algorithm
Com
puting t
ime(s
)
Fig. 9 Comparisons of computing time between the three different
methods for the broken-tooth signal.
5. Conclusions
The paper aims to provide guidance on choosing the
appropriate amplitude of the added white noise for the
EEMD method and reduce the tremendous time waste
occurring in the EEMD method. To solve those problems,
based on the CEEMD method, the ICEEMD method is
addressed in this paper. Besides, the numerical examples
and the experimental examples have tested the ability of
the ICEEMD method. The comparisons with the EEMD
and CEEMD methods show that the ICEEMD method
outperforms the EEMD method in alleviating the mode
mixing and splitting as well as reducing the time waste
and also outperforms the CEEMD method in alleviating
the mode mixing and splitting. This paper indicates that
the ICEEMD method is seemingly an effective and
efficient method for gearbox fault signal processing. In
addition, combined with some other methods[13, 14], the
ICEEMD method may achieve better results in analyzing
gearbox faulty signals.
Acknowledgments
The author thanks Kayvan J. Rafiee for his generous
providing free gearbox vibration dataset on his personal
official webpage. The work is supported by Natural
Science Fund of Shandong Province (ZR2012EEL07),
Development Program of Science and Technology of
Weifang City (201104050, 201104063) and Young
Science Fund of Weifang University (2012Z20).
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Jinshan Lin received his BE degree from Shandong University
in Mechanical Engineering and Automation in 2000. Next, he received his ME degree from University of Jinan in Mechanical and Electronic Engineering in 2006. Since 2006, he have taught at Weifang University. Currently, his research interests include fault detection, pattern classification and signal processing.
IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012 ISSN (Online): 1694-0814 www.IJCSI.org 199
Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.